Need for Matching

Part 1 · Impedance Matching

Key point: maximum power is at \(R_L = R_S\). A bigger \(V_L\) is not more power.

\(V_S = 2\) V peak unless a step says otherwise. Green line is the match. Green band is still at least 90% of \(P_{\mathrm{avail}}\).

1
Matched, 50 Ω.
\(R_S = R_L = 50\). \(V_L\) is 1.000 V. \(P_L\) and the power in \(R_S\) are both 10.00 mW. Fraction is 100.0%. The dot sits on the peak.
2
\(R_L\) too small.
10 Ω load. \(V_L\) falls to 0.333 V. \(P_L\) is 5.56 mW, 55.6% of available. Mismatch loss is 2.55 dB. Status says undermatched.
3
Bigger voltage, less power.
\(R_L = 350~\Omega\). \(V_L\) is 1.750 V, higher than the match. \(P_L\) is only 4.38 mW, 43.8%. Overmatched.
4
50 Ω is a choice.
75 Ω source into 75 Ω. Still matched, \(V_L\) still 1.000 V, but \(P_L\) is 6.67 mW. The green peak moved with \(R_S\).
5
Same match, bigger source.
Still 75 Ω into 75 Ω. \(V_S\) is 4 V. \(V_L\) is 2.000 V and \(P_L\) is 26.67 mW. The peak did not move. Power scaled with \(V_S^2\).

Resistive ports only. A real load is \(Z = R + jX\). Part 2 is that number, and how frequency moves \(X\).

Power into \(R_L\)

Fixed source \(R_S\) and open-circuit \(V_S\) · vary the load · log \(R_L\)

\(R_S\) (Ω) \(R_L\) (Ω) \(V_S\) (V peak)

Source V_S behind R_S (blue) into load R_L (rose, to ground). Operating point is V_L.

\(P_L\) vs \(R_L\) · green band ≈ 90% of available · red dot = this load

Status—

V_load—

P_load—

P in R_S—

P_available—

Fraction delivered—

Mismatch loss—

The three power equations

\[ V_L = V_S\,\dfrac{R_L}{R_S + R_L} \]

= —

\[ P_L = \dfrac{1}{2}\,\dfrac{|V_L|^2}{R_L} \]

= —

\[ P_{\mathrm{avail}} = \dfrac{V_S^2}{8\,R_S} \]

= —

Peak source \(V_S\); average powers. Curve is blue near the match and red when \(P_L < 90\%\) of available.